# Integration 2.pdf - 5.4 The Fundamental Theorem of Calculus...

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94. (The integral’s value is about 0.693.) In Exercises 95–102, use a CAS to perform the following steps: a. Plot the functions over the given interval. 1 b. Partition the interval into 200, and 1000 subinter- vals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). L 1 L p 1 5.4 The Fundamental Theorem of Calculus 325 d. Solve the equation for x using the av- erage value calculated in part (c) for the partitioning. 98. 1 x d d ] ] 5.4 The Fundamental Theorem of Calculus In this section we present the Fundamental Theorem of Calculus, which is the central theorem of integral calculus. It connects integration and differentiation, enabling us to compute inte- grals using an antiderivative of the integrand function rather than by taking limits of Riemann sums as we did in Section 5.3. Leibniz and Newton exploited this relationship and started mathematical developments that fueled the scientific revolution for the next 200 years. Along the way, we present an integral version of the Mean Value Theorem, which is another important theorem of integral calculus and is used to prove the Fundamental Theorem. Mean Value Theorem for Definite Integrals In the previous section we defined the average value of a continuous function over a closed interval [ a , b ] as the definite integral divided by the length or width of the interval. The Mean Value Theorem for Definite Integrals asserts that this average value is always taken on at least once by the function ƒ in the interval. The graph in Figure 5.16 shows a positive continuous function defined over the interval [ a , b ]. Geometrically, the Mean Value Theorem says that there is a number c in [ a , b ] such that the rectangle with height equal to the average value ƒ( c ) of the function and base width has exactly the same area as the region beneath the graph of ƒ from a to b . b - a y = ƒ s x d b - a 1 b a ƒ s x d dx H ISTORICAL B IOGRAPHY Sir Isaac Newton (1642–1727) FIGURE 5.16 The value ƒ( c ) in the Mean Value Theorem is, in a sense, the average (or mean ) height of ƒ on [ a , b ]. When the area of the rectangle is the area under the graph of ƒ from a to b , ƒ s c ds b - a d = L b a ƒ s x d dx . ƒ Ú 0, y x a b 0 c y f ( x ) f ( c ), b a average height THEOREM 3— The Mean Value Theorem for Definite Integrals If ƒ is continu- ous on [ a , b ], then at some point c in [ a , b ], ƒ s c d = 1 b - a L b a ƒ s x d dx . Proof If we divide both sides of the Max-Min Inequality (Table 5.4, Rule 6) by we obtain min ƒ 1 b - a L b a ƒ s x d dx max ƒ.